Understanding a crude estimate for the number of lattice points inside a ball

Question

I've been reading the appendix (A lattice sum) of this write up by Keith Conrad and pretty much understand most of the argument but I'm stuck on the following:

Let $S(x)$ denote the number of non-zero lattice points inside the ball $\{\mathbf{x} \in \mathbf{R}^d: ||\mathbf{x}|| \leq \sqrt{x} \}$ Since a $d$-dimensional ball of radius $r$ has volume $C_r r^d$ we have, in this case, that $S(x) \leq B_d (\sqrt{x})^d $ for some constant $B_d$ only dependent on $d$.

However he also says that, it follows from this that there exists positive constants $A_d$ and $B_d$ such that $$A_d (\sqrt{x})^d \leq S(x)\leq B_d (\sqrt{x})^d$$ for large $x$. I understand the upper bound as one cannot have more lattice points than a balls volume obviously but I'm not sure how he established the lower bound. This seems like an important principal that could be applied to other problems so I'd really like to understand it.

Thanks.

Answer 1

There exists two constants $a_d$ and $b_d$ (both positive) such that $a_d\|x\|_\infty \leq \|x\| \leq b_d \|x\|_\infty.$ Then $S(r) \subset S_\infty(r/a_d)$ and $S_\infty(r) \subset S(b_d r).$ It is clear (because they are products of integer intervals) that $|S_\infty(r)| = [r]^d.$ The result follows.